Question 256355
with continuous compounding formula, you get:


{{{f = p * e^rt}}}


f = future value
p = present amount
r = annual interest rate
t = number of years


the e is the symbol for the scientific constant of 2.718281828...


This is an irrational number (never ending non repeating fractional part).


It's called Napier's constant.


the formula for the base of e to an exponent is {{{e^x}}} where x is the exponent.


the inverse formula of {{{e^x}}} is equal to {{{ln(x)}}} which means the natural log of x.


the basis definition of {{{e^x}}} is:


{{{y = e^x}}} if and only if {{{x = ln(y)}}}.


this is nothing more than your normal logarithm functions except you have a base of e.


It's no different than the statement:


{{{y = 10^x}}} if and only if {{{log(10,x) = y}}}


for example:


let x = 3


{y = 10^3}}} = 10*10*10 = 1000


the statement becomes:


{{{y = 10^3}}} if and only if {{{log(10,1000) = 3}}}.


you can confirm that {{{log(10,1000) = 3}}} by using your calculator and taking the log of 3.   


you will get 1000.


the base e is nothing more than another base to work with.


In your problem, here's how you would work it.


formula is {{{f = p * e^rt}}}.


you are given:


p = $7,700
r = 8.78%
t = 10 years


first thing you need to do is take all the dollar signs and commas out of p to get:


p = 7700


next thing you need to do is divide 8.78% by 100% to get .0878.


in the formula, you need to work with the rate, not the percent.


since r is an innual interest rate already, you do not need to adjust it any further.


since t is already specified in years, you do not need to adjust it any further.


plug these values into your formula to get:


{{{f = p * e^rt}}} becomes


{{{f = 7700 * e^(.0878*10)}}}


simplify to get:


{{{f = 7700 * e^(.878)}}}


use your calculator to get {{{e^(.878)}}} = 2.406082726


alternatively, you can substitute the constant of 2.718281828 to get:


{{{2.718281828^(.878) = 2.406082725.


It's a little off from what the calculater would give you but it's very close.   That's beceuase of some rounding going on when you see the display of the number.   The internally stored number is accurate to more decimal places.


your equation becomes:


{{{f = 7700 * 2.406082726}}}


solve for f to get:


f = 18526.83699


that's equivalent to $18,526.84.


for some good examples of continuous compounding, check out the web.


do a search on "continuous compounding".


one such website is <a href = "http://moneychimp.com/articles/finworks/continuous_compounding.htm" target = "_blank">http://moneychimp.com/articles/finworks/continuous_compounding.htm</a>


It has a calculator that lets you calculate different numbers and see the effect of continuous compounding compared to discrete compounding such as yearly, monthly, daily, and hourly.


It also has a box where you can put in the number of time periods you want.